On an invariant measure for homeomorphisms of a circle with a point of break
type: Item
instance of: scholarly article
title: On an invariant measure for homeomorphisms of a circle with a point of break
zbMATH Open document ID: 0921.58035
DOI: 10.1007/BF02463336
author: K. Khanin
author: A. A. Dzhalilov
publication date: 1999-09-30T00:00:00Z
review text: The authors study homeomorphisms of the circle \(S^1=[0,1)\) with a simple point of break, that is, with a jump of the first derivative at a point. As is well known in the case of generic rotation numbers, the invariant measure for a \(C^{2+\nu}\)-diffeomorphism is absolutely continuous with respect to the Lebesgue measure. The authors show that the situation becomes diametrically opposite in the presence of a single point of break-type singularity, namely, they prove that an invariant measure is always singular with respect to the Lebesgue measure.
Mathematics Subject Classification ID: 37A99
zbMATH DE Number: 1292496
zbMATH Keywords: simple point of break
zbMATH Keywords: rotation numbers
zbMATH Keywords: invariant measure
zbMATH Keywords: break-type singularity
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
cites work: Sur les courbes definies par les équations différentielles à la surface du tore
cites work: Q5520417
cites work: Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. (On smooth conjugacy of diffeomorphisms of the circle with rotations)
cites work: Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne
cites work: A new proof of M. Herman's theorem
cites work: Q4711018
cites work: Smooth conjugacy and renormalisation for diffeomorphisms of the circle
cites work: Q3359437
cites work: Universal estimates for critical circle mappings
full work available at URL: https://doi.org/10.1007/bf02463336
OpenAlex ID: W2040470280
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 1292496